Thermalization and dynamics in the single impurity Anderson model

We analyze the process of thermalization, dynamics and the eigenstate thermalization hypothesis (ETH) for the single impurity Anderson model, focusing on the Kondo regime. For this we construct the complete eigenbasis of the Hamiltonian using the numerical renormalization group (NRG) method in the language of the matrix product states. It is a peculiarity of the NRG that while the Wilson chain is supposed to describe a macroscopic bath, very few single particle excitations already suffice to essentially thermalize the impurity system at finite temperature, which amounts to having added a macroscopic amount of energy. Thus given an initial state of the system such as the ground state together with microscopic excitations, we calculate the spectral function of the impurity using the microcanonical and diagonal and grand canonical ensembles. By adding or removing particles at a certain Wilson energy shell on top of the ground state, we find qualitative agreement between the spectral functions calculated for different ensembles. This indicates that the system thermalizes in the long-time limit, and can be described by an appropriate statistical-mechanical ensemble. Moreover, by calculating the impurity spectral density at the Fermi level and the dot occupancy for energy eigenstates relevant for microcanonical ensemble, we find good support for ETH. The ultimate mechanism responsible for this effective thermalization within the NRG can be identified as Anderson orthogonality: the more charge that needs to flow to or from infinity after applying a local excitation within the Wilson chain, the more the system looks thermal afterwards at an increased temperature. For the same reason, however, thermalization fails if charge rearrangement after the excitation remains mostly local.